Transition Models

Transition Models Peter Midford University of Kansas

Modeling Languages Representation Focused Languages Controlled Vocabulary Conventional Languages (C, Java) (AML) (Xrate)

Stake Holders • Users • Use Cases • Phylogenetic inference • Identify Sites in target protein • Identify Variability • Develop models • Education • DataMiners • Application Developers

Legacy Modeling Language • Describe • Models • optimization criteria • constraints • Most focus on operations research domains

Legacy Modeling Language examples • GAMS • Inclusion of sets • Free form syntax • MPL • Pascal-like syntax • MML • Physiology focus • Java-like syntax • AMPL • AIMMS

MPL example { Planning.mpl } { Aggregate production planning for 12 months } TITLE Production_Planning; INDEX product = 1..3; month = (January,February,March,April,May,June,July, August,September,October,November,December); DATA price[product] := (105.09, 234.00, 800.00); Demand[month,product] := 1000 DATAFILE(demand.dat); ProductionCapacity[product] := 1000 (10, 42, 14); ProductionCost[product] := (64.30, 188.10, 653.20); InventoryCost := 8.8 ; DECISION VARIABLES Inventory[product,month] -> Invt Production[product,month] -> Prod Sales[product,month] -> Sale MACRO Revenues := SUM(product,month: price * Sales); TotalCost := SUM(product,month: InventoryCost * Inventory + ProductionCost * Production); MODEL MAX Profit = Revenues - TotalCost ; SUBJECT TO InventoryBalance[product,month] -> IBal : Inventory = Inventory[month-1] + Production - Sales ; BOUNDS Sales < Demand ; Production < ProductionCapacity ; Inventory[month=January..November] < 90000 ; Inventory[month=December] = 20000 ; END

MML Example // introductory ODE model math main { realDomain t; t.min=0; t.max=4; t.delta=0.1; real rate = 1; real u(t); // declaration of u when (t=t.min) u=1; // initial condition for u u:t = -rate*u; // ODE state equation for u }

MML Example (2) // constraint example math main { real a, b, c; a = b + 1; // eqn 1 c = a + b; // eqn 2 a = 5; // eqn 3 }

XRate • An implementation of Phylogrammars • Generalization of • Hidden Markov Models • Stochastic Context Free Grammars • Lisp Syntax • Concise • Besides point substitution, also handles feature prediction • May have some difficulty exactly expressing models in other frameworks. • Example (Kimura 2 Parameter!)

Kimura 2 parameter Example ;; The grammar. ;; For Kimura's two-parameter model, the concept of a grammar is a bit superfluous, ;; but necessary "boilerplate code" to do this sort of thing in xgram. (grammar (name KimuraTwoParameterModel) ;; Transformation rules. These follow the pattern for a null model with rate matrix X. ;; There is one emit state, corresponding to emissions from X. (transform (from (S)) (to (X S*))) ;; A hacky (but common) way of conditioning on the observed alignment length ;; is to set both transition probabilities from the emit state to one: (transform (from (S*)) (to (S)) (prob 1)) (transform (from (S*)) (to ()) (prob 1)) ;; Finally we clear a flag, indicating we don't want to re-estimate the rule probabilities by EM. (update-rules 0) ;; Here are the parameters for Kimura's model. (params ((alpha .4)) ;; transition rate ((beta .1)) ;; transversion rate ) ;; end params ;; Now here is the algebraic structure of the rate matrix. (chain ;; The state of this chain is a single symbol from alphabet DNA. ;; Call this symbol X. (terminal (X)) ;; The following line indicates that the initial probabilities and mutation rates ;; should be treated as fixed parametric functions, not free variables. (update-policy parametric) ;; initial probability distribution (initial (state (a)) (prob 0.25)) (initial (state (c)) (prob 0.25)) (initial (state (g)) (prob 0.25)) (initial (state (t)) (prob 0.25))

Kimura 2 parameter Example (continued) ;; mutation rates (mutate (from (a)) (to (c)) (rate beta)) (mutate (from (a)) (to (g)) (rate alpha)) (mutate (from (a)) (to (t)) (rate beta)) (mutate (from (c)) (to (a)) (rate beta)) (mutate (from (c)) (to (g)) (rate beta)) (mutate (from (c)) (to (t)) (rate alpha)) (mutate (from (g)) (to (a)) (rate alpha)) (mutate (from (g)) (to (c)) (rate beta)) (mutate (from (g)) (to (t)) (rate beta)) (mutate (from (t)) (to (a)) (rate beta)) (mutate (from (t)) (to (c)) (rate alpha)) (mutate (from (t)) (to (g)) (rate beta)) ) ;; end chain X ) ;; end grammar ;; Standard DNA alphabet (alphabet (name DNA) (token (a c g t)) (complement (t g c a)) (extend (to n) (from a) (from c) (from g) (from t)) (extend (to x) (from a) (from c) (from g) (from t)) (extend (to u) (from t)) (extend (to r) (from a) (from g)) (extend (to y) (from c) (from t)) (extend (to m) (from a) (from c)) (extend (to k) (from g) (from t)) (extend (to s) (from c) (from g)) (extend (to w) (from a) (from t)) (extend (to h) (from a) (from c) (from t)) (extend (to b) (from c) (from g) (from t)) (extend (to v) (from a) (from c) (from g)) (extend (to d) (from a) (from g) (from t)) (wildcard *) ) ;; end alphabet DNA

AML • XML schema for describing Modeling languages • XSL stylesheets exist for GAMS, AMPL, MPL • Avoids use of MathML (for ease of validation) • PDF is available, but homepage for schema seems to have disappeared.

AML Language Elements • Sets • Parameters • Variables • Macros • Objective • Constraints

Simple OWL Ontology for Transition Models • Class hierarchy based on complexity • Other relations to define common components of models • Parameters • Constants • Constraints • Portions have been added to Nexml schema

Extensibility • Ontology will include definitions of the standard models as well as scaffolding for describing new models • An application with a new type of model can extend the existing vocabulary in a derivative ontology • If the new type of model, or variants of it, become popular, 'some authority' could add it to the standard, base model ontology

Transition Models